Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2015, Article ID 573146, 6 pages
http://dx.doi.org/10.1155/2015/573146
Research Article
Global Stability of an Anthrax Model with Environmental
Decontamination and Time Delay
Steady Mushayabasa
Department of Mathematics, University of Zimbabwe, P.O. Box MP 167, Harare, Zimbabwe
Correspondence should be addressed to Steady Mushayabasa; [email protected]
Received 17 March 2015; Accepted 21 June 2015
Academic Editor: Mauro Sodini
Copyright © 2015 Steady Mushayabasa. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Anthrax occurs worldwide and is associated with sudden death of cattle and sheep. This paper considers an epidemic model
of anthrax in animal population, only. The susceptible animal is assumed to be infected, only, through ingestion of the disease
causing pathogens. The proposed model incorporates time delay and environmental decontamination by humans. The time delay
represents the period an infected animal needs to succumb to anthrax-related death. By constructing suitable Lyapunov functionals,
we demonstrate that the global dynamics of this model fully hinges on whether the associated reproductive number is greater or less
than unity. The effectiveness of environmental decontamination on eradication of anthrax in the community is explored through
the reproductive number.
1. Introduction
Anthrax is an acute zoonotic disease caused by the sporeforming bacterium Bacillus anthracis [1]. The disease can
infect all warm-blooded animals, including humans. However, ruminants, particularly cattle and sheep, are more
susceptible [2]. The disease is associated with death of
cattle and sheep, such that very few livestock producers
or veterinaries have witnessed the disease or its signs [2].
Outbreaks often occur when livestock are grazing on neutral
or slightly alkaline soils and have been exposed to the spores
through ingestion of contaminated soil in endemic areas or
forage is sparse because of overgrazing or drought or when
soil has been disturbed by digging or other human activities
[2]. Anthrax is one of the most dramatic diseases affecting
wild animal in Africa. For instance, within Etosha National
Park, anthrax has been attributed to the death of a variety of
herbivores, from elephants to ostriches [3].
The use of mathematical models to explore the spread and
control of infectious diseases has proved to be an important
tool for the scientists and epidemiologists. In 1983, Hahn and
Furniss [4] constructed a deterministic model to explore the
impact of environmental contamination and animal carcasses
on driving anthrax epidemic. Within their framework they
assumed that upon infection infected animals die immediately and there are no births or deaths from causes other than
anthrax related [4]. More, recently, Friedman and Yakubu
[3] extended the model of Hahn and Furniss [4] to study
the effects of anthrax transmission, carcass ingestion, carcass
induced environmental contamination, and migration rates
on the persistence or extinction of animal population. Their
work revealed among others that decreasing the levels of
carcass ingestion by removal of carcasses in game reserves
may not always lead to a reduction in anthrax cases.
Although anthrax infection in cattle is regarded often as
a fatal disease with no signs [2], biologically the infection
should take a period before an infected animal succumbs
to anthrax-induced mortality, and this period may play an
important role in controlling anthrax outbreaks. So incorporating this reason we extend the work of Hahn and Furniss [4]
to include a fixed delay. Apart from the delay, our new model
incorporates the role of human effort on decontamination of
the environment through the destruction of animal carcass
and disinfection of the ground or area that contained the
carcass by adding lime. The main goal of this paper is to
explore the effectiveness of environmental decontamination
on controlling anthrax outbreaks and to study the global
stability of the anthrax model with time delay.
2
Discrete Dynamics in Nature and Society
2. Model Formulation and Analysis
A framework to assess the dynamics of anthrax in livestock
is proposed. Hahn and Furniss’s [4] deterministic model
of anthrax dynamics provides the starting point for our
discussion on the effectiveness of environmental decontamination on controlling the spread of anthrax. The framework
is governed by the following system of nonlinear ordinary
differential equations:
(1)
𝑑𝑃
= 𝛼𝐶 (𝑡) − (𝜙 + 𝜖) 𝑃 (𝑡) .
𝑑𝑡
𝛼𝑏
≤
},
(𝜙 + 𝜖) 𝜔
−
0
(3)
𝐴=(
− (𝜙 + 𝑑)
𝛼
𝛽𝑏
𝜇 ).
− (𝜙 + 𝜖)
(4)
The trace and determinant of 𝐴 are, respectively, given by
The first equation describes the dynamics of the susceptible
animals. The parameter 𝑏 denotes the entrance of new
animals through birth and they are assumed to be susceptible
to the disease; 𝛽 is the disease transmission rate which occurs
when a susceptible animal ingests the free-living spores while
grazing in an endemic area; 𝜇 denote the permanent exit
of the animals due to natural causes or other reasons not
related to anthrax infection. Upon infection, we assume that
an infected animal dies without displaying clinical signs of the
disease. Thus, the second equation captures the dynamics of
the carcasses of animals that may have succumbed to anthraxinduced death; 𝑑 denote the decay rate of carcasses which
is assumed to be constant; 𝜙 denote the rate of disinfection
or decontamination of infected areas through the removal
or destruction of animal carcasses and adding of lime to
the ground where a decomposing animal carcass would have
been identified. The third equation describes dynamics of
free-living spores or pathogens; 𝛼 denote the rate at which
carcasses of animals that would have died of anthrax and not
properly destroyed shed the bacteria into the environment;
𝜖−1 is the average life span of free-living pathogens. The length
of survival of anthrax free-living spores in the environment is
estimated to be around 200 years [5]; thus 𝜖 = 1/(365×200) =
0.000014 day−1 .
For biological reasons we will study the dynamics of
system (1) in the closed set
Γ = {(𝑆, 𝐶, 𝑃) ∈ R3+ : 0 ≤ 𝑆 + 𝐶 ≤
𝛽𝑏
𝜇
𝛽𝑏 ) .
𝐽=(
0 − (𝜙 + 𝑑)
𝜇
0
𝛼
− (𝜙 + 𝜖)
−𝜇
It is clear that −𝜇 is an eigenvalue of matrix 𝐽. The other two
are determined from matrix 𝐴. Consider
𝑑𝑆
= 𝑏 − 𝛽𝑃 (𝑡) 𝑆 (𝑡) − 𝜇𝑆 (𝑡) ,
𝑑𝑡
𝑑𝐶
= 𝛽𝑃 (𝑡) 𝑆 (𝑡) − (𝜙 + 𝑑) 𝐶 (𝑡) ,
𝑑𝑡
an infection-free equilibrium given by E0 = (𝑏/𝜇, 0, 0). The
Jacobian matrix of system (1) evaluated about E0 is
𝑏
, 0≤𝑃
𝜔
tr (𝐴) = − (2𝜙 + 𝜖 + 𝑑) ,
det (𝐴) = (𝜙 + 𝑑) (𝜖 + 𝜙) [1 −
𝛽𝛼𝑏
].
𝜇 (𝜙 + 𝑑) (𝜙 + 𝜖)
(5)
Let the reproductive number of system (1) be
𝛽𝛼𝑏
.
𝜇 (𝜙 + 𝑑) (𝜙 + 𝜖)
R𝑎 =
(6)
Since tr(𝐴) < 0 and det(𝐴) > 0 (when R𝑎 < 1) it follows
that the equilibrium point E0 is locally asymptotically stable
if and only if R𝑎 < 1.
Biologically, the term R𝑎 represents the average number
of new anthrax cases generated when susceptible animals
ingest disease causing pathogens. It gives a measure of the
power of anthrax to invade the cattle population in the
presence of environmental decontamination.
Direct calculation can easily show that system (1) has two
possible equilibrium points, namely, the infection-free E0
and the endemic equilibrium point E∗ , given by
E∗ = {𝑆∗ =
𝜇 (𝜖 + 𝜙)
𝑏
, 𝐶∗ =
(R𝑎 − 1) , 𝑃∗
𝜇R𝑎
𝛼𝛽
𝜇
= (R𝑎 − 1)} .
𝛽
(7)
From (7) it is evident that E∗ makes biological sense whenever R𝑎 > 1.
(2)
where R3+ denotes the nonnegative cone of R3 including its
lower dimensional faces and 𝜔 ≤ min(𝜇, 𝜙, 𝑑). It can easily be
verified that Γ is positively invariant with respect to (1).
2.1. Equilibria and the Reproductive Number. In the absence
of anthrax infection in the community, system (1) admits
2.2. Effectiveness of Environmental Decontamination. In this
section we explore the strength of environmental decontamination on reducing or eliminating new anthrax infections.
In the absence of environmental decontamination in the
community (𝜙 = 0) the average number of new anthrax
infections generated through spores ingestion is modeled by
the threshold quantity R0 which is given by
R0 =
𝛽𝛼𝑏
.
𝜇𝜖𝑑
(8)
Discrete Dynamics in Nature and Society
3
System (10) satisfies the following initial conditions:
100
E(𝜙)
80
𝑆 (𝜃) = 𝜙1 (𝜃) ,
60
40
𝐶 (𝜃) = 𝜙2 (𝜃) ,
20
𝑃 (𝜃) = 𝜙3 (𝜃) ,
0
0
0.005
0.01
0.015
𝜙
0.02
0.025
To explore the effectiveness of environmental decontamination on controlling new anthrax infections we define the
efficacy function
𝜖𝑑
] × 100%.
= [1 −
(𝜖 + 𝜙) (𝜙 + 𝑑)
(9)
2.3. Anthrax Model with Time Delay. Although anthrax
infection in cattle is often a fatal disease with no clinical
signs displayed by an infected animal, it is worth noting that
the infection should take a period before an infected animal
succumbs to anthrax-induced death and the size of this
period may play an important role in controlling the outbreak
of this disease. So incorporating this reason we introduce a
time delay into system (1) to represent the aforementioned
period:
𝑑𝑆
= 𝑏 − 𝛽𝑃 (𝑡) 𝑆 (𝑡) − 𝜇𝑆 (𝑡) ,
𝑑𝑡
𝑑𝑃
= 𝛼𝐶 (𝑡) − (𝜙 + 𝜖) 𝑃 (𝑡) .
𝑑𝑡
where 𝜙𝑖 (𝜃) for 𝑖 = 1, 2, 3 denote the nonnegative continuous
functions on 𝜃 ∈ [−𝜏, 0]. All the parameters in system (10)
are the same as in system (1) except for the positive constant
𝜏 which represents the length of the delay. It can be verified
that Γ is positively invariant with respect to system (10).
We denote by 𝜕Γ and Γ̂ the boundary and the interior of
Γ in R3+ , respectively. With the same motivation as before,
we introduce the reproductive number of differential-delay
model (10) which is given by a similar expression
R𝑎 =
From (9) it is clear that the effectiveness of environmental
decontamination depends on the length of survival of freeliving pathogens in the environment and the decay rate
of undestroyed carcasses of animals that have succumbed
to anthrax infection. In Figure 1 we numerically explore
the effectiveness of different environmental decontamination
levels.
Numerical results in Figure 1 suggest that environmental
decontamination rate 0.05 day−1 can be effective in reducing
the generation of new anthrax infections by 50%. Further, we
note that environmental decontamination rate greater than or
equal to 0.01 day−1 can be effective in attaining 100% control
on the spread of anthrax in the community.
𝑑𝐶
= 𝛽𝑃 (𝑡 − 𝜏) 𝑆 (𝑡 − 𝜏) − (𝜙 + 𝑑) 𝐶 (𝑡) ,
𝑑𝑡
𝜃 ∈ [−𝜏, 0] ,
0.03
Figure 1: Graphical illustration showing the strength of environmental decontamination on controlling the spread of anthrax. We
consider 𝜖 = 0.00014 day−1 [5] and 𝑑 = 0.06 day−1 [4].
R − R𝑎
] × 100%
𝐸 (𝜙) = [ 0
R0
(11)
𝛽𝛼𝑏
.
𝜇 (𝜙 + 𝑑) (𝜙 + 𝜖)
(12)
2.3.1. Anthrax-Free Equilibrium Point and Its Stability. Direct
calculation shows that system (10) has the same disease-free
equilibrium E0 = (𝑏/𝜇, 0, 0) as in system (1). The Jacobian
matrix of system (10) about E0 is
−𝜇
𝐽𝑤 = (
0
0 − (𝜙 + 𝑑)
0
𝛼
−
𝛽𝑏
𝜇
𝛽𝑏 −𝜆𝜏 ) .
𝑒
𝜇
− (𝜙 + 𝜖)
(13)
From (13) it follows that the characteristic equation is
(𝜆 + 𝜇) [𝜆2 + (2𝜙 + 𝜖 + 𝑑) 𝜆 + (𝜖 + 𝜙) (𝜙 + 𝑑)
𝛽𝛼𝑏 −𝜆𝜏
−
𝑒 ] = 0.
𝜇
(14)
Since 𝜆 = −𝜇 < 0 is a root of (14), we only need to consider
𝜆2 + 𝑎11 𝜆 + 𝑎12 + 𝑏11 𝑒−𝜆𝜏 = 0,
(15)
where
𝑎11 = (2𝜙 + 𝜖 + 𝑑) > 0,
𝑎12 = (𝜖 + 𝜙) (𝑑 + 𝜙) > 0,
(16)
𝛽𝛼𝑏
𝑏11 = −
< 0.
𝜇
Equation (14) with 𝜏 = 0 is
(10)
𝜆2 + 𝑎11 𝜆 + 𝑎12 + 𝑏11 = 0.
(17)
If 𝜏 = 0, we have already deduced that all roots of system (15)
have negative real parts when R𝑎 < 1, and only one root of
(15) has positive real part when R𝑎 > 1.
4
Discrete Dynamics in Nature and Society
For 𝜏 > 0, we assume that 𝜆 = 𝑖𝜔 (𝜔 > 0) is a root of (15).
This is the case if and only if
2
− 𝜔 + 𝑖𝑎11 𝜔 + 𝑎12 + 𝑏11 (cos 𝜔𝜏 − 𝑖 sin 𝜔𝜏) = 0.
(18)
The derivative of 𝑉 along the solutions of (10) is given by
𝑑𝑈 (𝑥𝑡 )
𝑑𝑡
Separating the real and imaginary parts yields
=𝛼
− 𝜔2 + 𝑎12 = − 𝑏11 cos 𝜔𝜏,
(19)
− 𝑎11 𝜔 = − 𝑏11 sin 𝜔𝜏.
+ (𝜙 + 𝑑)
(20)
≤ (𝜙 + 𝑑) (𝜖 + 𝜙) [
Let 𝑧 = 𝜔2 , so that (21) reduces to
2
2
2
𝑧2 + (𝑎11
− 2𝑎12 ) 𝑧 + 𝑎12
− 𝑏11
= 0.
(21)
Substituting for 𝑎11 , 𝑎12 , and 𝑏11 gives
2
2
𝑧2 + (𝜖 + 𝜙) (𝜙 + 𝑑) 𝑧
2
𝛽𝛼𝑏
+ (𝜖 + 𝜙) (𝜙 + 𝑑) [1 − (
) ] (22)
𝜇 (𝜖 + 𝜇) (𝜙 + 𝑑)
2
2
𝑑𝑃 (𝑡)
𝑑𝑡
(25)
= 𝛽𝛼𝑃 (𝑡) 𝑆 (𝑡) − (𝜙 + 𝑑) (𝜙 + 𝜖) 𝑃 (𝑡)
Adding up the squares of both equations, we obtain
2
2
2
𝜔4 + (𝑎11
− 2𝑎12 ) 𝜔2 + 𝑎12
− 𝑏11
= 0.
𝑡
𝑑𝐶 (𝑡) 𝑑
+ (𝛼𝛽 ∫ 𝑃 (𝜃) 𝑆 (𝜃) 𝑑𝜃)
𝑑𝑡
𝑑𝑡
𝑡−𝜏
𝛽𝛼𝑏
− 1] 𝑃 (𝑡)
𝜇 (𝜙 + 𝑑) (𝜖 + 𝜙)
= (𝜙 + 𝑑) (𝜖 + 𝜙) (R𝑎 − 1) 𝑃 (𝑡) ≤ 0.
Therefore, 𝑈̇ ≤ 0 holds for all 𝑃 ≥ 0 and R𝑎 ≤ 1.
Furthermore, 𝑈̇ = 0 if and only if 𝑃 = 0, or R𝑎 = 1. Therefore,
the largest compact invariant set in {(𝑆, 𝐶, 𝑃)} ∈ Γ : 𝑈̇ = 0,
when R𝑎 ≤ 1, is the singleton {E0 }. Thus, LaSalle’s invariance
principle [7] implies that E0 is globally asymptotically stable
in Γ. This proves the theorem.
2.3.2. Endemic Equilibrium Point and Its Stability
= 0.
Theorem 3. Suppose R𝑎 > 1; then the equilibrium point E0
is unstable.
Substituting R𝑎 into (22) gives
2
2
2
2
𝑧2 + (𝜖 + 𝜙) (𝜙 + 𝑑) 𝑧 + (𝜖 + 𝜙) (𝜙 + 𝑑) [1 − R2𝑎 ]
= 0.
(23)
Whenever R𝑎 < 1, (23) has two roots which have a positive
product implying that they are complex or they are real but
they have the same sign. In addition, they have negative
sum which implies that they are either real and negative or
complex conjugates with negative real parts. Consequently,
(23) does not have positive real roots which lead to the
conclusion that there is no 𝜔 such that 𝑖𝜔 is a solution of (14).
Therefore, it follows from Lemma 2.4 of Ruan and Wei [6]
that the real parts of all eigenvalues of characteristic equation
(14) are negative for all values of the delay 𝜏 ≥ 0. Thus, we
have the following theorem.
Proof. From the discussion in Section 2.3.1, we have deduced
that the characteristic equation associated with E0 is given
by (14), and we only need to consider (15). From the above
computations it can easily be verified that if 𝜏 = 0, (15) has a
positive root when R𝑎 > 1. Now, let 𝑖𝜔 (𝜔 > 0) be a root of
(15). Then solving (23) gives
𝜔+2 =
[
⋅ [−1 + √ 1 − 4
[
𝜔−2 =
0
Theorem 1. The equilibrium point E of system (10) is locally
asymptotically stable for all time delay 𝜏 ≥ 0 if R𝑎 < 1.
In the next theorem we establish the global stability of the
infection-free equilibrium for system (10).
Theorem 2. The equilibrium point E0 of system (10) is globally
asymptotically stable when R𝑎 ≤ 1 for all 𝜏 ≥ 0.
Proof. We denote by 𝑥𝑡 the translation of the solution of
system (10); that is, 𝑥𝑡 = (𝑆(𝑡 + 𝜃), 𝐶(𝑡 + 𝜃), 𝑃(𝑡 + 𝜃)), where
𝜃 ∈ [−𝜏, 0]. Let us define a Lyapunov functional
𝑈 (𝑥𝑡 ) = 𝛼𝐶 (𝑡) + 𝛼𝛽 ∫
𝑡
𝑡−𝜏
+ (𝜙 + 𝑑) 𝑃 (𝑡) .
𝑃 (𝜃) 𝑆 (𝜃) 𝑑𝜃
(24)
1
2
2
(𝜖 + 𝜙) (𝜙 + 𝑑)
2
(1 − R2𝑎 )
2
(𝜖 + 𝜙) (𝜙 + 𝑑)
]
,
2]
(26)
]
1
2
2
(𝜖 + 𝜙) (𝜙 + 𝑑)
2
[
⋅ [−1 − √ 1 − 4
(1 − R2𝑎 )
2
]
.
2]
(27)
(𝜖 + 𝜙) (𝜙 + 𝑑)
[
]
For R𝑎 > 1 it is evident that (26) makes sense while (27) is
meaningless.
Define
𝜏𝑗 =
𝜇 (2𝜙 + 𝜖 + 𝑑) 𝜔+
1
[arcsin (−
) + 2𝑛𝜋] ,
𝜔+
𝛽𝛼𝑏
(28)
𝑗 = 0, 1, 2, . . . .
Then (14) has a pair of purely imaginary roots ±𝜔 when 𝜏 = 𝜏𝑗
and has no roots appearing on the imaginary axis when 𝜏 ≠ 𝜏𝑗
for 𝑗 = 0, 1, . . . .
Discrete Dynamics in Nature and Society
5
Further, let 𝜆(𝜏) = 𝜓(𝜏) + 𝑖𝜔(𝜏) be the root of (14)
satisfying 𝜓(𝜏𝑗 ) = 0 and 𝜔(𝜏𝑗 ) = 𝜔+ .
Differentiating both sides of (14) with respect to 𝜏 we get
(2𝜆 + 𝑎11 − 𝑏11 𝜏𝑒−𝜆𝜏 )
𝑑𝜆
= 𝑏11 𝜏𝑒−𝜆𝜏 .
𝑑𝜏
Separating the real and imaginary parts we have
𝜔3 − 𝑎22 𝜔 = − 𝑏21 𝜔 cos 𝜔𝜏 + 𝑏22 sin 𝜔𝜏,
𝑎21 𝜔2 − 𝑎23 = − 𝑏21 𝜔 sin 𝜔𝜏 − 𝑏22 cos 𝜔𝜏.
(29)
This gives
(36)
Squaring and adding the two equations of (36) yields
(
−1
𝑑𝜆
)
𝑑𝜏
=
2𝜆 + 𝑎11 𝜏
−
𝑏11 𝜆𝑒−𝜆𝜏 𝜆
(30)
−𝜇 (𝜖 + 2 (𝜆 + 𝜙) + 𝑑) 𝜆𝜏 𝜏
=
𝑒 − .
𝛽𝛼𝑏
𝜆
2
2
2
2
𝜔6 + (𝑎21
− 2𝑎22 ) 𝜔4 + (𝑎22
− 2𝑎21 𝑎23 − 𝑏21
) 𝜔2 + 𝑎23
2
− 𝑏22
= 0,
(37)
where
Thus
sign {
𝑑 (Re𝜆 (𝜏𝑗 ))
𝑑𝜏
= sign {
} = sign {Re (
2
2
2
− 2𝑎21 𝑎23 − 𝑏21
= 𝜇2 (𝜖 + 𝑑)2 R2𝑎 + 2𝜇 (𝜖 + 𝜙)
𝑎22
𝑗
𝜇2 (2𝜔+2 + 2𝜙2 + 𝜖2 + 𝑑2 + 2𝜙 (𝜖 + 𝑑))
(𝛽𝛼𝑏)
2
}
(31)
By applying Lemma 2.4 in Ruan and Wei [6] and observing
that (15) has a positive real root when 𝜏 = 0, we obtain that
characteristic equation (14) has a positive root at least for all
𝜏 ≥ 0. Therefore, the equilibrium point E0 of system (10) is
unstable when R𝑎 > 1. This completes the proof.
Now, we investigate the effect of the time delay on the
local stability of E∗ . The characteristic equation of system (10)
at the endemic equilibrium E∗ takes the following form:
(32)
where
𝑎21 = 2𝜙 + 𝜖 + 𝑑 + 𝜇R𝑎 ,
When 𝜏 = 0, (32) becomes
= (cos 𝜔𝜏 − 𝑖 sin 𝜔𝜏) (𝑖𝑏21 𝜔 + 𝑏22 ) .
> 0.
Hence, if R𝑎 > 1, (32) has no positive roots. Accordingly,
if R𝑎 > 1, the endemic equilibrium E∗ exists and is locally
asymptotically stable for all 𝜏 ≥ 0. Further, using a Lyapunov
functional, we can obtain the following theorem.
Theorem 4. Whenever R𝑎 > 1, then the unique equilibrium
E∗ of system (10) is globally asymptotically stable in Γ̂ for all
𝜏 ≥ 0.
= 𝑆 (𝑡) − 𝑆∗ ln 𝑆 (𝑡) + 𝐶 (𝑡) − 𝐶∗ ln 𝐶 (𝑡)
𝛽𝑃∗ 𝑆∗
(𝑃 (𝑡) − 𝑃∗ ln 𝑃 (𝑡))
𝛼𝐶∗
+∫
𝑡
𝑡−𝜏
(34)
By the Hurwitz criteria, all the roots of (34) have only negative
real parts. Thus E∗ is locally asymptotically stable when 𝜏 = 0.
Now, we consider the case 𝜏 ≠ 0. If 𝑖𝜔 (𝜔 > 0) is a solution of
(32), we have
− 𝑖𝜔3 − 𝑎21 𝜔2 + 𝑖𝑎22 𝜔 + 𝑎23
2
2
2
𝑎23
− 𝑏22
= 𝜇2 (𝜙 + 𝑑) (𝜖 + 𝜙) (R𝑎 − 1) (3R𝑎 − 1)
+
𝑏22 = 𝜇 (𝜙 + 𝑑) (𝜖 + 𝜙) R𝑎 .
+ 𝜇 (𝜙 + 𝑑) (𝜖 + 𝜙) (R𝑎 − 1) = 0.
2
(38)
𝑉𝑎 (𝑥𝑡 )
(33)
𝑏21 = (𝜙 + 𝑑) (𝜖 + 𝜙) ,
𝜆3 + 𝑎21 𝜆2 + 𝜇 (2𝜙 + 𝜖 + 𝑑) R𝑎 𝜆
𝜇
− 1)] > 0,
(2𝜙 + 𝜖 + 𝑑)
Proof. We consider the following Lyapunov functional:
𝑎22 = (2𝜙 + 𝜖 + 𝑑) 𝜇R𝑎 + (𝜙 + 𝑑) (𝜖 + 𝜙) ,
𝑎23 = 𝜇 (𝜙 + 𝑑) (𝜖 + 𝜙) (2R𝑎 − 1) ,
⋅ (𝜙 + 𝑑) (2𝜙 + 𝜖 + 𝑑)
⋅ [1 + R𝑎 (
> 0.
𝜆3 + 𝑎21 𝜆2 + 𝑎22 𝜆 + 𝑎23 = 𝑒−𝜆𝜏 (𝑏21 𝜆 + 𝑏22 ) ,
2
2
− 2𝑎22 = (𝜙 + 𝑑) + (𝜖 + 𝜙) + 𝜇2 R2𝑎 > 0,
𝑎21
𝑑𝜆 −1
) }
𝑑𝜏
𝜏=𝜏
(35)
(39)
𝛽 [𝑃 (𝜃) 𝑆 (𝜃) − 𝑃∗ 𝑆∗ ln 𝑃 (𝜃) 𝑆 (𝜃)] 𝑑𝜃.
Differentiating 𝑉𝑎 along the solution (𝑆, 𝐶, 𝑃) of system (10)
and using the identities
𝑏 = 𝛽𝑃∗ 𝑆∗ + 𝜇𝑆∗ ,
(𝜙 + 𝑑) =
𝛽𝑃∗ 𝑆∗
,
𝐶∗
(𝜙 + 𝜖) =
𝛼𝐶∗
,
𝑃∗
(40)
6
Discrete Dynamics in Nature and Society
one gets
𝑑𝑉𝑎
𝑆 (𝑡)
𝑆∗
𝑆∗
) + 𝛽𝑃∗ 𝑆∗ (3 −
= 𝜇𝑆∗ (2 − ∗ −
𝑑𝑡
𝑆
𝑆 (𝑡)
𝑆 (𝑡)
−
𝐶 (𝑡) 𝑃∗ 𝑆 (𝑡 − 𝜏) 𝑃 (𝑡 − 𝜏) 𝐶∗
−
𝐶∗ 𝑃 (𝑡)
𝑆∗ 𝐶 (𝑡) 𝑃∗
+ ln
(41)
𝑆 (𝑡 − 𝜏) 𝑃 (𝑡 − 𝜏)
).
𝑆 (𝑡) 𝑃 (𝑡)
Here,
2≤
𝑆∗
𝑆 (𝑡)
+
,
𝑆∗
𝑆 (𝑡)
(42)
for all 𝑆(𝑡) ≥ 0, because the arithmetic mean is greater than or
equal to the geometric mean. Further, note that 3 − 𝑎 − 𝑏 − 𝑐 +
ln(𝑎𝑏𝑐) ≤ 0 for any 𝑎 > 0, 𝑏 > 0, and 𝑐 > 0; and the equality
is satisfied if and only if 𝑎 = 𝑏 = 𝑐 = 1, and, consequently,
𝑉𝑎̇ (𝑡) ≤ 0. Moreover, the largest invariant set of 𝑉𝑎̇ (𝑡) = 0 is a
singleton where 𝑆(𝑡) ≡ 𝑆∗ , 𝐶(𝑡) = 𝐶∗ , and 𝑃(𝑡) ≡ 𝑃∗ . By the
Lyapunov-LaSalle invariance principle [7], we obtain global
asymptotic stability of the endemic equilibrium (𝑆∗ , 𝐶∗ , 𝑃∗ )
under the condition R𝑎 > 1.
3. Concluding Remarks
Anthrax epidemic is now recognized among other factors
as the leading cause of species extinctions. In this paper,
we propose and analyze an anthrax epidemic model. The
model focuses on anthrax transmission in animal population
only. Our model is based on the assumption that the disease
transmission occurs only when a susceptible animal ingests
the disease causing pathogen from contaminated soil in
endemic areas when forage is sparse because of overgrazing
or drought or when soil has been disturbed by digging or
excavations. The proposed model incorporates time delay
and environmental decontamination effort. The time delay
represents the period that is needed for an infected animal
to succumb to anthrax-induced death. We determine the
reproductive number and establish that the global dynamics
depends on whether the reproductive number is greater than
one.
Conflict of Interests
The author declares no conflict of interests.
Acknowledgments
The author is grateful to the anonymous referee and handling
editor for their valuable comments and suggestions.
References
[1] CDC, “Use of anthrax vaccine in the United States: recommendations of the Advisory Committee on Immunization Practices
(ACIP), 2009,” Morbidity and Mortality Weekly Report, vol. 59,
no. 6, pp. 1–30, 2010, http://www.cdc.gov/mmwr/.
[2] A. N. Survely, B. Kvasnicka, and R. Torell, “Anthrax: a guide for
livestock producers,” Cattle Producer’s Library CL613, Western
Beef Resource Committe, 2006.
[3] A. Friedman and A. A. Yakubu, “Anthrax epizootic and migration: persistence or extinction,” Mathematical Biosciences, vol.
241, no. 1, pp. 137–144, 2013.
[4] B. D. Hahn and P. R. Furniss, “A deterministic model of an
anthrax epizootic: threshold results,” Ecological Modelling, vol.
20, no. 2-3, pp. 233–241, 1983.
[5] S. S. Lewerin, M. Elvander, T. Westermark et al., “Anthrax
outbreak in a Swedish beef cattle herd—1st case in 27 years: case
report,” Acta Veterinaria Scandinavica, vol. 52, no. 1, article 7,
2010.
[6] S. Ruan and J. Wei, “On the zeros of transcendental functions
with applications to stability of delay differential equations with
twon delays,” Dynamics of Continuous, Discrete and Impulsive
Systems, vol. 10, pp. 863–874, 2003.
[7] J. S. LaSalle, The Stability of Dynamical Systems, vol. 25 of CBMSNSF Regional Conference Series in Applied Mathematics, SIAM,
Philadelphia, Pa, USA, 1976.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Mathematical Problems
in Engineering
Journal of
Mathematics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Discrete Dynamics in
Nature and Society
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Abstract and
Applied Analysis
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014